The part (2) seems more subtle and has been treated by many authors. For weak solutions, the non-uniqueness results have been obtained by V. Scheffer ([1]}), A. Shnirelman ([2]}, [3]}) and Camillo De Lellis, László Székelyhidi ([4]}, [5]}). In particular, a great progress in the construction of Hölder continuous solution was made by Camillo De Lellis, László Székelyhidi etc in recent years. In fact, Camillo De Lellis and László Székelyhidi developed an iterative scheme in [6]}, together with the aid of Beltrami flow on \(T^3\)  and Geometric Lemma, and constructed a continuous periodic solution which satisfies the prescribed kinetic energy. The solution is a superposition of infinitely many weakly interacting Beltrami flows. Building on the iterative techniques in [6]} and Nash-Moser mollify techniques, they constructed Hölder continuous periodic solutions with exponent \(\theta <\frac{1}{10}\) , which satisfies the prescribed kinetic energy in [8]}. P. Isett in [9]} constructed Hölder continuous periodic solutions with any \(\theta <\frac{1}{5}\) , and the solution has compact support in time. By introducing some new devices in [10]}, Camillo De Lellis, László Székelyhidi and T. Buckmaster constructed Hölder continuous weak solutions with \(\theta <\frac{1}{5}\) , which satisfies the prescribed kinetic energy, also see [11]}.
In \(R^3\) , P. Isett and Sung-jin Oh in [12]} constructed Hölder continuous solutions with \(\theta <\frac{1}{5}\) , which satisfies the prescribed kinetic energy or is a perturbation of smooth Euler flow. Recently, S. Daneri obtained dissipative Hölder solutions for the Cauchy problem of incompressible Euler flow in [13]}.
